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Sometimes I rush a Zonya's Hourglass if I need some survivability, switch out the Void Staff for an Abyssal Sceptre if the enemy team is not so tanky, and I'll replace Sorceror's shoes for Merc Treads if I get a full build with a Void staff.

Runes:

I use Flat armor and Magic resist Seals and Glyphs, Flat Ability power Quintessences, and Magic Penetration Marks.

The Armor seals can be switched out for health per level because of the damage reduction from Shunpo but I prefer the armor to deal with the enemy ADs.

Sometimes I rush a Zonya's Hourglass if I need some survivability, switch out the Void Staff for an Abyssal Sceptre if the enemy team is not so tanky, and I'll replace Sorceror's shoes for Merc Treads if I get a full build with a Void staff.

Runes:

I use Flat armor and Magic resist Seals and Glyphs, Flat Ability power Quintessences, and Magic Penetration Marks.

The Armor seals can be switched out for health per level because of the damage reduction from Shunpo but I prefer the armor to deal with the enemy ADs.

why Hex first? its pretty expensive compared to the DFG. would it be any worse if i went DFG first then hex 2nd? because i usually start off with a kages pick for gold and AP early game which builds into the DFG

as far as katarinas go, I would consider myself a pretty good one. runes are a pretty touchy topic with her, I run two different pages for her, one that is tankier, and one that is straight AP and MR pen. But for masteries, def go 21/9/0 getting the MR in the def tree.

Her build should start with boots+3pots

My first back is usually kages pick+sorc boots (prioritize boots because the spell pen will do more dmg than more ap, and you will move faster)

depending on how you do in lane, and who you are laning against, you will want to rush one of two things,
neg cloak (tanky mid, or if you are losing to a strong mid)
or finish DFG (if you are facerolling)
usually after I will build the other after I finish each one.

This neg cloak will become an abyssal scepter very quickly.

What many kats don't know, is that spell pen is actually stronger in some situations than actual AP because of her ratios.

Then I will build a revolver if I need the spell vamp, or a needlessly large rod if I am stupidly fed.

The revolver can stay as a revolver as you build a deathcap, because you really want gunblade for its active, but I find you need the extra AP over that active that you can't use because it will cancel your ult is more useful.

some other good items are rylais, and then maybe hourglass. You should build hourglass if you need specifically armor, but I usually buy rylais for the slow+ult combo.

As the game goes on, just build more AP and tanky items. I wouldn't recommend void staff though because that is just a straight bursty item, and you could put a better item there that will let you actually put out your damage in the middle of a teamfight. But to each their own, if you find something is working, go for it.

Katarina is one of my best and most played champs.. over the past 5-6 months I've changed my play style a fair bit.

Currently I have two rune, mastery and build routes, both of which work well, though my current preferred route is less conventional.

Route 1. - the more conventional one
runes: MP reds, Movement quints (or AP if you so desire.. personally I love Mov on katarina). hp/level yellows, ap/level blues.
masteries.
21/9/0 - pretty standard AP offensive + hp from the defense tree build.
Skill route (lvl 1-4): Q > W > E > Q
lvl5-18 priority (R, then Q, then W, then E)
Item route:
boots + 3pots
First B. hextech (+ sorc boots if you can last till 2k gold instead of 1200)
Then depending on how well you and your team are doing, either go straight for a rabadons (if doing good) or go for rylai's/Abyssal first for the defensive side,
Finish the hextech if their team is full of escapes/chasers, otherwise leave it till last
rest is situational.
=======================
Route 2 - Less conventional, but very productive
runes. flat AD red, AD quints (or movement if you want. I go AD as this build is more tanky). flat MR yellow and blue
Masteries: 9/21/0 (http://leaguecraft.com/masteries/1300040001000000003014321000330010000000000000000) - The idea behind this mastery + runepage combo is that as a mid you're going to be against an AP, so with all this bonus MR and damage reduction you will be VERY tanky. The minion damage reduction + passive 2 reduction realy adds up defensively when you shunpo onto their AP and take a few hits from their minions.
The 3hp/5 also adds up a lot surprisingly. with this build I don't go for an early hextech, 3 pots is plenty to play aggressive, take hits and still stay in lane till 1500-2k gold.

The AD runes are beneficial as it allows you to last hit easier and hit harder with your basic attacks vs their AP mid who will stack MR instead of Armour. Also your W and R scale with AD well too.

item route:
boots + pots
Aim for at least 1550 gold before going back
first b. Sorc boots + kagi's lucky pick
second b aim for when you have 2-2.2k gold. haunting Guise (1400ish) + a base item for abyssal scepter (or hextech IF you feel you really need it.. I have rarely needed to myself)
3rd b. Abyssal scepter + start on your rabadons
to aim for in the long run. Rabadons + deathfire grasp + situationals

The concept of this build is to deny mid as much as possible. be aggressive to them but don't push the lane so that you deny them gold and exp. the reason for the 3x Spell Pen items is that you will end up dealing true damage to everyone who doesn't build MR, and near true damage to any who build some (with tanks being the exception). Even though you've got AD runes, your abilities convert that bonus AD into magical damage so you're still gaining the full bonus from them on your spam (W) and ult (R) which you should be saving for 2-6 seconds into a team fight so that all the CC are blown allowing you to clean up.

I've been very successful with this build and even find champs who I used to have issues with A LOT easier now (Mordkeiser, Diana, LeBlanc for example) due to how much less damage they deal. The loss in the later offensive masteries is made up for the most part by the high spell pen and higher survivability.

Well I main Katarina and have watched pretty much every TiensiNoAkuma and Scarra videos around on her. Their are many different kind of builds for her some less used then others (Reginald and Dx30's builds) But Ill give a few I guess.

I highly suggest running the following runes:
Ms Quints
Health per level Glyphs (extra health goes a long way)
Flat Magic Pen Reds (Increases damage throughout the entire game)
Mr per level seals (Since you will be playing mid lane)

As for masteries it is up to you what to run myself I have 3 mastery pages for Katarina depending on who I am going up against mid. Most commonly though people use 21/9/0.

As far as item builds go anyone who tells you to build Hextech Gunblade or Rylais on the new Katarina isn't really correct. Rushing Gunblade on the old Kat was ideal because it gave you the slow to keep the opponent in your ultimate and your Q scaled off the AD it gave however with the new Katarina her Q dosent scale of AD anymore and her ult is shorter therefore Gunblade becomes quite useless besides the spell vamp (which could be evened out with a few pots). As far for Rylais goes it dosent fit Katarina since Rylais is for someone who spams spells and sticks to people (Ryze, Akali, Vlad) Katarina unleashes it in quick bursts. Whats the point of slowing someone without killing them? As Tiensi says if your picking up Rylais your saying to your teammates "Here! Let me slow them for you and you guys go ahead and kill them!" Your playing an assassin you want to dish out a ton of damage and use your passive to move on.

So for items using Tiensi's build (My favorite)

You start boots 3
Go into Sorc Shoes and Kage Pick
Then you immediately rush Deathcap (Aim to get it before 20 minutes, if you get it at 15 your snowballing hard and will usually win)
Once you get Deathcap your at the strongest point in your game and should capatalize on it and force fights.
After Deathcap you either go Abyssal, Zyhonias or finish your DeathFirst (depending on the situation)
Your end build should look something like
Abyssal, Zyhonias, Deathcap, DFG, Void Stagg and Guardian Angel.

Scarra usually builds the same thing however rushes Abyssal first for a better lanning phase.

As for Dx30's build.

He will start boots 3
Go into boots of mobility and Kages pick
Pick up a Haunting Guise
Then follow Tiensi's build more or less depending on the situation.

If you need more help on Katarina I highly recommend watching TiensiNoAkuma's videos on youtube. My skills with Kat went through the roof after watching his.

As for Skill Orders I like Q->W->E and R whenever its up however others like their own methods. I like maxing Q as it optimizes 1v1 damage output because of it's proc and it allows me to farm from a nicer distance vs W where I have to walk within its range and W I feel Ill surely take harass most of the time.

Also something you should not forget is Kat only really becomes useful around lvl 4-8 depending on how experienced you are with her. At that point you should try to roam whenever your lane is pushed or take wraiths. However until you get your deathcap aim to farm extremely hard and snowball once you get the Deathcap if you can before then odds are you will carry your team to victory

I highly suggest running the following runes:
Ms Quints
Health per level Glyphs (extra health goes a long way)
Flat Magic Pen Reds (Increases damage throughout the entire game)
Mr per level seals (Since you will be playing mid lane)

I believe you mean mr/level glypths and hp/level yellows. The way you said gives you about half the mr and hp overall that you could achieve

1) this post will ppb get downvoted. (just warning in advance, almost all my Kat posts tend to get downvoted into oblivion)
2) You can check ANY of my claims.
3) This post doesn't apply to jungle Katarina or top lane tank Kat.

a) AD is a wasted stat on her. Pretty much all it's good for is lasthitting. People think AD is good on her because she has AD scalings, but then you do the math and find out that every action she can perform in the game scales way harder with AP than AD. (Keep in mind that gold-efficiëncy wise AD is 2ce as expensive as AP.)
b) Spellvamp is barely worth it. No, seriously, it isn't. The reason it's still bought is because gunblade was core on old Kat for several reasons that changed during the rework. The largest change to the spellvamp component is the fact that new BB is considered and AoE spell. This dramatically lessens the effect of spellvamp on Katarina compared to the old days.
c) Gunblade isn't worth it. Reasons 1 and 2 contribute to this. The only extra thing Gblade has is the activatable, the slow on which isn't all too useful anymore since the nerf to her ultimate. If you REALLY want an activatable damage nuke, DFG offers a stronger one. (inb4 I get comments on how "cutting a second off of her ult is a buff" again, please note that her old ult dealt slightly more damage after 2 seconds than the new one does. Do the math, don't forget to incorporate Killer Instincts.)
d) Don't invest in CDR. It's a bad stat on her due to Voracity.
e) You should focus on proccing voracity as much and as fast as possible. This is Kat's main damage amplification tool, and it's the main thing that separates her from other champions. WIth new Kat, this pretty much means getting AP/Mpen is your primary goal, as damage is the only thing you offer to your team. The secondary thing you need is defence, cause living longer means more damage.
f) DO NOT get Rylai. It was a bad item on old Kat, and it got nerfed even harder on new Kat. It's just not worth the cost. You're better off just getting a giant's belt and flat AP if you really need the tankyness. As for the slow: it wasn't overly useful on old Kat due to her being a full out burst assassin rather than a "stick to a target and chase" assassin (like Akali is more or less); and it's even worse on new Kat due to the BB changes.

Summarizing: Invest in pure raw damage as much as you can muster. Sorc shoes are the only boots that really make sense. As for items, AP vs Mpen is a rough discussion and it's situational, as with any champion.

Items
Boots + 3 Health pots
Kage's Lucky Pick on first back (at least)
Get wards and potions as needed (probably try and hold onto one ward for using with E to escape)
Sorcerer's Shoes
Hextech Revolver (if you need the sustain)
Negatron into Abyssal Scepter
Upgrade to Deathfire Grasp
Rabadon's Deathcap

At this point you should have: DFG, Sorc's, Abyssal, Deathcap, and possibly Revolver.

For your last remaining items, consider purchasing: Voidstaff for more Mpen, Zhonya's for more AP and its active, GA for defense, and upgrading Revolver to Gunblade/Will of the Ancients (depending on team comp).